Completing the Square for Vertex Form - Math 2

1. A quadratic function $f(x) = x^2 - 8x + 19$ models the height of a projectile, where $x$ represents time in seconds and $f(x)$ represents height in feet. To find when the projectile reaches its minimum height, Maria completes the square. What is the vertex form of this function, and what does the vertex represent in this context?
   Answer: $f(x) = (x - 4)^2 + 3$; the projectile reaches its minimum height of 3 feet at 4 seconds To complete the square for $f(x) = x^2 - 8x + 19$: Take half of the coefficient of $x$ (which is $-8/2 = -4$), square it to get 16, then add and subtract 16: $f(x) = x^2 - 8x + 16 + 19 - 16 = (x - 4)^2 + 3$. The vertex is at $(4, 3)$, meaning the minimum height of 3 feet occurs at 4 seconds. Choice B incorrectly keeps 19 as the constant term. Choice C has the wrong sign in the binomial. Choice D uses the wrong value from completing the square process.
2. A quadratic function has the form $f(x) = 3x^2 - 18x + 35$. A student attempts to complete the square but makes an error in the process. Which of the following correctly shows the vertex form and identifies the vertex?
   Answer: $f(x) = 3(x - 3)^2 + 8$; vertex at $(3, 8)$ For $f(x) = 3x^2 - 18x + 35$, factor out 3: $f(x) = 3(x^2 - 6x) + 35$. Complete the square: $(-6/2)^2 = 9$. So $f(x) = 3(x^2 - 6x + 9 - 9) + 35 = 3((x-3)^2 - 9) + 35 = 3(x-3)^2 - 27 + 35 = 3(x-3)^2 + 8$. The vertex is $(3, 8)$. Choice B has wrong sign for the $x$-coordinate. Choice C forgets to distribute the 3 through the subtracted 9. Choice D incorrectly uses 6 instead of 3 as the $x$-coordinate of the vertex.
3. The function $g(x) = 4x^2 - 16x + 19$ is rewritten in vertex form. If this function represents the distance of a moving object from a reference point, where $x$ is time in seconds and $g(x)$ is distance in meters, at what time is the object closest to the reference point, and what is this minimum distance?
   Answer: Closest at $t = 2$ seconds with minimum distance of 3 meters When you see a quadratic function representing distance over time, you're looking for the vertex of the parabola, which gives you the minimum distance since the coefficient of $x^2$ is positive. To find the vertex, you need to complete the square to convert $g(x) = 4x^2 - 16x + 19$ to vertex form. First, factor out the coefficient of $x^2$ from the first two terms: $g(x) = 4(x^2 - 4x) + 19$. To complete the square inside the parentheses, take half of the coefficient of $x$ and square it: $(-4/2)^2 = 4$. Add and subtract this inside: $g(x) = 4(x^2 - 4x + 4 - 4) + 19 = 4((x-2)^2 - 4) + 19 = 4(x-2)^2 - 16 + 19 = 4(x-2)^2 + 3$. The vertex form $g(x) = 4(x-2)^2 + 3$ shows the vertex is at $(2, 3)$, meaning the object is closest to the reference point at $t = 2$ seconds with a minimum distance of 3 meters. Looking at the wrong answers: (A) incorrectly identifies the time as 4 seconds, possibly confusing the coefficient inside the parentheses with the vertex location. (C) uses the correct time but gives 19 meters as the minimum distance, which is actually the y-intercept, not the vertex. (D) uses 16 seconds, likely misinterpreting the coefficient of $x$ in the original equation as the time. Remember: for vertex form $a(x-h)^2 + k$, the vertex is at $(h, k)$, and completing the square is your most reliable method to find it.
4. A student attempts to rewrite $g(x) = 2x^2 + 12x + 7$ in vertex form by completing the square. They write: $g(x) = 2(x^2 + 6x) + 7 = 2(x + 3)^2 + 7$. What error did the student make, and what is the correct vertex form?
   Answer: They forgot to subtract $2 \cdot 9 = 18$ when completing the square; correct form is $g(x) = 2(x + 3)^2 - 11$ When completing the square for $g(x) = 2x^2 + 12x + 7$, first factor out 2: $g(x) = 2(x^2 + 6x) + 7$. To complete the square inside the parentheses, add and subtract $(6/2)^2 = 9$: $g(x) = 2(x^2 + 6x + 9 - 9) + 7 = 2((x + 3)^2 - 9) + 7 = 2(x + 3)^2 - 18 + 7 = 2(x + 3)^2 - 11$. The student forgot to account for the fact that when 9 is inside the parentheses with the factor of 2, it becomes $2 \cdot 9 = 18$ that must be subtracted outside. Choices B, C, and D represent different types of errors but don't address the specific mistake made.
5. A quadratic function in the form $f(x) = x^2 + bx + c$ has vertex form $f(x) = (x + 4)^2 - 9$. If a student needs to verify this conversion by expanding the vertex form back to standard form, what should they get for the values of $b$ and $c$?
   Answer: $b = 8$ and $c = 7$ When you encounter questions about converting between standard form and vertex form of quadratic functions, you need to master the expansion process using the perfect square trinomial pattern. To verify the conversion, expand the vertex form $(x + 4)^2 - 9$ back to standard form. Start with the perfect square: $(x + 4)^2 = x^2 + 8x + 16$. This follows the pattern $(a + b)^2 = a^2 + 2ab + b^2$, where $a = x$ and $b = 4$, giving you $x^2 + 2(x)(4) + 4^2 = x^2 + 8x + 16$. Now subtract 9: $f(x) = x^2 + 8x + 16 - 9 = x^2 + 8x + 7$. Therefore, $b = 8$ and $c = 7$. Choice A gives $b = -4$ and $c = -9$, which comes from incorrectly taking coefficients directly from the vertex form without expanding. The $-4$ represents the x-coordinate of the vertex, not the coefficient of $x$ in standard form. Choice C gives $b = 4$ and $c = 16$, showing a sign error in $b$ (forgetting that $2 \times 4 = 8$, not 4) and failure to subtract the 9. Choice D gives $b = 8$ and $c = 16$, correctly finding $b$ but forgetting to subtract 9 from the constant term after expanding. Remember: when expanding $(x + h)^2$, the coefficient of $x$ in standard form is always $2h$, and you must carefully handle any constants added or subtracted outside the squared term.
6. The profit $P$ (in thousands of dollars) for a small business is modeled by $P(x) = -2x^2 + 16x - 30$, where $x$ is the number of years after 2020. A consultant needs to determine when the business will achieve maximum profit. After completing the square, what form should be used, and what business recommendation follows?
   Answer: $P(x) = -2(x - 4)^2 + 2$; maximum profit of $2,000 occurs in 2024 Starting with $P(x) = -2x^2 + 16x - 30$, factor out -2: $P(x) = -2(x^2 - 8x) - 30$. Complete the square: take half of -8, square it: $(-8/2)^2 = 16$. Add and subtract: $P(x) = -2(x^2 - 8x + 16 - 16) - 30 = -2((x-4)^2 - 16) - 30 = -2(x-4)^2 + 32 - 30 = -2(x-4)^2 + 2$. The vertex $(4, 2)$ means maximum profit of $2,000 occurs 4 years after 2020, which is 2024. Choice B has wrong sign. Choice C fails to distribute the -2 correctly. Choice D uses wrong vertex $x$-coordinate.
7. A quadratic function modeling population growth is given by $P(t) = -t^2 + 14t + 40$, where $P$ is population (in hundreds) and $t$ is years after 2020. A demographer needs to determine when maximum population occurs and what that maximum will be. What does completing the square reveal?
   Answer: $P(t) = -(t - 7)^2 + 89$; maximum population of 8,900 in year 2027 When you encounter a quadratic function in vertex form problems, you're looking to rewrite $ax^2 + bx + c$ as $a(x - h)^2 + k$, where the vertex is at $(h, k)$. For population models with a negative leading coefficient, the vertex represents the maximum point. To complete the square for $P(t) = -t^2 + 14t + 40$, first factor out the negative from the first two terms: $P(t) = -(t^2 - 14t) + 40$. Take half of the coefficient of $t$ (which is $-14$), square it: $(-14/2)^2 = 49$. Add and subtract this inside the parentheses: $P(t) = -(t^2 - 14t + 49 - 49) + 40 = -(t^2 - 14t + 49) + 49 + 40 = -(t - 7)^2 + 89$. This vertex form shows the maximum occurs at $t = 7$ with a population value of $89$. Since $t$ represents years after 2020, this means year $2020 + 7 = 2027$. The population is in hundreds, so $89 × 100 = 8,900$ people. Choice A incorrectly has $(t + 7)^2$, which would place the vertex at $t = -7$ (year 2013). Choice C correctly identifies the year but uses the wrong $k$-value of 40 instead of 89, missing the constant term arithmetic. Choice D places the vertex at $t = 14$ (year 2034), confusing the coefficient $-14$ with the vertex location. Remember: in vertex form $a(x - h)^2 + k$, the vertex is at $(h, k)$, and when $a < 0$, you have a maximum at that point.
8. A company's monthly revenue $R$ (in thousands) is modeled by $R(p) = -4p^2 + 32p - 60$, where $p$ is the price per unit in dollars. To maximize revenue, the company needs the vertex form. What price should they charge, and what will be the maximum monthly revenue?
   Answer: $R(p) = -4(p - 4)^2 + 4$; charge $4 per unit for maximum revenue of $4,000 When you encounter a quadratic revenue function, you need to convert it to vertex form to find the maximum revenue and optimal price. The vertex form $f(x) = a(x - h)^2 + k$ immediately shows the vertex at $(h, k)$, where $h$ is the price that maximizes revenue and $k$ is the maximum revenue. To convert $R(p) = -4p^2 + 32p - 60$ to vertex form, complete the square. First, factor out the coefficient of $p^2$ from the first two terms: $R(p) = -4(p^2 - 8p) - 60$. To complete the square inside the parentheses, take half of the coefficient of $p$ (which is $-8$), square it: $(-8/2)^2 = 16$. Add and subtract this inside: $R(p) = -4(p^2 - 8p + 16 - 16) - 60 = -4((p - 4)^2 - 16) - 60$. Distribute the $-4$: $R(p) = -4(p - 4)^2 + 64 - 60 = -4(p - 4)^2 + 4$. The vertex is at $(4, 4)$, meaning charge $4 per unit for maximum revenue of $4,000 (remember the revenue is in thousands). Choice A incorrectly identifies the optimal price as $8 instead of $4. Choice B has the wrong sign in the vertex form, showing $(p + 4)^2$ instead of $(p - 4)^2$. Choice C fails to complete the square properly, showing $-60$ as the $k$-value instead of $+4$. Study tip: Always double-check your completed square by expanding it back to standard form. This catches sign errors and ensures your vertex coordinates are correct.
9. A water fountain shoots water in a parabolic arc given by $h(t) = -16t^2 + 24t + 4$, where $h$ is height in feet and $t$ is time in seconds. To design a decorative element at the water's peak, what is the vertex form and when does the peak occur?
   Answer: $h(t) = -16(t - 0.75)^2 + 13$; peak height of 13 feet at 0.75 seconds When you encounter a parabolic function in standard form, converting to vertex form reveals the maximum or minimum point directly. The vertex form is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex coordinates. To convert $h(t) = -16t^2 + 24t + 4$ to vertex form, you'll complete the square. First, factor out the coefficient of $t^2$ from the first two terms: $h(t) = -16(t^2 - 1.5t) + 4$. To complete the square inside the parentheses, take half of the coefficient of $t$ (which is $-1.5$), square it: $(-1.5 ÷ 2)^2 = (-0.75)^2 = 0.5625$. Add and subtract this inside the parentheses: $h(t) = -16(t^2 - 1.5t + 0.5625 - 0.5625) + 4 = -16((t - 0.75)^2 - 0.5625) + 4$. Distributing: $h(t) = -16(t - 0.75)^2 + 16(0.5625) + 4 = -16(t - 0.75)^2 + 9 + 4 = -16(t - 0.75)^2 + 13$. This gives us vertex $(0.75, 13)$, meaning the peak occurs at $t = 0.75$ seconds with height 13 feet. Answer B is correct. Answer A incorrectly uses $(t + 0.75)$, which would place the vertex at $t = -0.75$ (before launch). Answer C has the right time but wrong peak height (4 instead of 13). Answer D doubles the time to 1.5 seconds, likely from confusing $-1.5t$ with the vertex location. Remember: in vertex form $a(x - h)^2 + k$, the vertex is at $(h, k)$, and the sign inside the parentheses is opposite to the $h$-coordinate.
10. A parabolic arch for a bridge has the equation $y = -0.5x^2 + 4x + 6$, where $x$ and $y$ are measured in meters. An engineer needs the vertex form to determine the arch's highest point and optimal support placement. What is the vertex form, and what is the practical meaning of the vertex coordinates?
    Answer: $y = -0.5(x - 4)^2 + 14$; highest point is 14 meters at horizontal position 4 meters For $y = -0.5x^2 + 4x + 6$, factor out -0.5: $y = -0.5(x^2 - 8x) + 6$. Complete the square: $(-8/2)^2 = 16$. So $y = -0.5(x^2 - 8x + 16 - 16) + 6 = -0.5((x-4)^2 - 16) + 6 = -0.5(x-4)^2 + 8 + 6 = -0.5(x-4)^2 + 14$. The vertex $(4, 14)$ means the highest point of the arch is 14 meters high, located 4 meters from the reference point horizontally. Choice B has incorrect sign. Choice C doesn't properly add the constant term. Choice D has calculation errors in both coordinates.
11. A farmer wants to fence a rectangular field using 200 feet of fencing, with one side against an existing barn (requiring no fence). If $w$ is the width perpendicular to the barn, the area is $A(w) = w(200 - 2w) = -2w^2 + 200w$. What is the vertex form, and what dimensions maximize the area?
    Answer: $A(w) = -2(w - 50)^2 + 5000$; width = 50 feet, length = 100 feet, maximum area = 5000 sq ft When you encounter optimization problems involving quadratic functions, you're looking for the maximum or minimum value by converting to vertex form: $f(x) = a(x - h)^2 + k$, where the vertex is at $(h, k)$. To convert $A(w) = -2w^2 + 200w$ to vertex form, complete the square. First, factor out the coefficient of $w^2$: $A(w) = -2(w^2 - 100w)$. To complete the square inside the parentheses, take half of the coefficient of $w$ and square it: $(-100 ÷ 2)^2 = 50^2 = 2500$. Add and subtract this inside: $A(w) = -2(w^2 - 100w + 2500 - 2500) = -2((w - 50)^2 - 2500) = -2(w - 50)^2 + 5000$. The vertex is at $(50, 5000)$, meaning maximum area occurs when $w = 50$ feet. Since the total fencing is 200 feet and covers width + width + length (one width is against the barn), we have $2w + \text{length} = 200$, so length $= 200 - 2(50) = 100$ feet. The maximum area is 5000 square feet. Choice A incorrectly places the vertex at $w = 100$, which would give length = 0 (impossible). Choice B has the wrong sign in the vertex form, placing the vertex at $w = -50$ (meaningless for width). Choice C has the correct vertex location but wrong maximum value, likely from calculation errors in completing the square. Remember: when completing the square, carefully track the algebra, especially when factoring out coefficients. The vertex of a parabola opening downward gives you the maximum value.
12. The area $A$ of a rectangular garden with a fixed perimeter of 40 feet can be expressed as $A(w) = w(20 - w) = -w^2 + 20w$, where $w$ is the width in feet. A gardener wants to find the dimensions that maximize the area. What is the completed square form, and what dimensions should be used?
    Answer: $A(w) = -(w - 10)^2 + 100$; width = 10 feet, length = 10 feet for maximum area of 100 sq ft Starting with $A(w) = -w^2 + 20w$, factor out -1: $A(w) = -(w^2 - 20w)$. Complete the square: $(20/2)^2 = 100$. So $A(w) = -(w^2 - 20w + 100 - 100) = -((w-10)^2 - 100) = -(w-10)^2 + 100$. The vertex $(10, 100)$ means maximum area of 100 sq ft when width = 10 feet. Since perimeter is 40 feet, length = $20 - 10 = 10$ feet. The optimal garden is square. Choice B has wrong sign. Choice C has wrong maximum value. Choice D gives impossible dimensions (length can't be 0).
13. The profit function for a small business is given by $P(x) = -2x^2 + 16x - 26$, where $x$ represents the number of items produced (in hundreds) and $P(x)$ represents profit in thousands of dollars. If the business owner completes the square to find the vertex form, which statement correctly describes the maximum profit scenario?
    Answer: Maximum profit of $6,000 occurs when producing 400 items Completing the square for $P(x) = -2x^2 + 16x - 26$: Factor out -2 from the first two terms: $P(x) = -2(x^2 - 8x) - 26$. Complete the square inside: $x^2 - 8x = (x - 4)^2 - 16$. So $P(x) = -2[(x - 4)^2 - 16] - 26 = -2(x - 4)^2 + 32 - 26 = -2(x - 4)^2 + 6$. The vertex is at $(4, 6)$. Since $x$ is in hundreds of items, this means 400 items, and since $P(x)$ is in thousands of dollars, this means $6,000 profit. Choice B uses 800 items (double). Choice C misinterprets the scale. Choice D uses 4,000 items (ten times too large).
14. The temperature $T$ (in degrees Fahrenheit) in a greenhouse $h$ hours after midnight follows $T(h) = 2h^2 - 16h + 98$. The greenhouse manager needs to know when the minimum temperature occurs and what that temperature is. What does completing the square reveal?
    Answer: $T(h) = 2(h - 4)^2 + 66$; minimum temperature of 66°F occurs at 4:00 AM When you encounter a quadratic function in vertex form questions, you're looking to rewrite the equation as $a(x-h)^2 + k$ where the vertex is at point $(h,k)$. Since the coefficient of $h^2$ is positive (2), this parabola opens upward, making the vertex the minimum point. To complete the square for $T(h) = 2h^2 - 16h + 98$, first factor out the coefficient from the first two terms: $T(h) = 2(h^2 - 8h) + 98$. To complete the square inside the parentheses, take half of the coefficient of $h$ (which is -8), square it: $(-8/2)^2 = 16$. Add and subtract this inside: $T(h) = 2(h^2 - 8h + 16 - 16) + 98 = 2((h-4)^2 - 16) + 98 = 2(h-4)^2 - 32 + 98 = 2(h-4)^2 + 66$. This reveals the vertex at $(4, 66)$, meaning the minimum temperature of 66°F occurs at $h = 4$ (4:00 AM). Answer A incorrectly keeps the constant as 98 instead of properly distributing the factored 2. Answer B has the wrong sign in the vertex form ($h + 4$ instead of $h - 4$), which would place the minimum at a negative time. Answer D incorrectly identifies the vertex's x-coordinate as 8 instead of 4, likely from confusing the coefficient -8 with the vertex location. Remember: when completing the square, carefully track your arithmetic, especially when factoring out coefficients and distributing them back through the expression.
15. The area of a rectangular garden is modeled by $A(w) = w^2 + 8w - 20$, where $w$ represents the width in feet and the length is $(w + 10)$ feet. By completing the square, determine the width that minimizes the area and interpret why this result might seem unusual in a real-world context.
    Answer: Width of -4 feet minimizes area to -36 square feet; this is unusual because negative dimensions and area are not physically meaningful When you encounter a quadratic function representing area in a real-world context, you need to find the vertex using completing the square, then carefully interpret whether the mathematical result makes physical sense. To complete the square for $A(w) = w^2 + 8w - 20$, you take half the coefficient of $w$ (which is 8), square it to get 16, then rewrite: $A(w) = (w^2 + 8w + 16) - 16 - 20 = (w + 4)^2 - 36$. The vertex form shows the minimum occurs at $w = -4$ with an area of $-36$ square feet. Choice A incorrectly identifies the width as 8 feet, likely confusing the coefficient 8 with the vertex location. The interpretation about maximizing versus minimizing area misses the real issue. Choice B gets the correct minimum area of -36 but incorrectly calculates the width as 4 feet, probably forgetting the negative sign when solving $(w + 4)^2 = 0$. The interpretation only identifies negative area as problematic. Choice C makes an error in calculating the minimum area as 20 instead of -36, possibly substituting incorrectly into the original function. Choice D correctly identifies that the width of -4 feet produces a minimum area of -36 square feet, and properly recognizes that both negative width and negative area are physically meaningless in this context. Study tip: When completing the square in real-world problems, always check if your mathematical answer makes physical sense. Negative dimensions, areas, or volumes should trigger you to reconsider the domain or recognize limitations of the mathematical model.
16. An engineer designs a suspension bridge cable following $y = 0.1x^2 - 2x + 15$, where $x$ is horizontal distance (in meters) from the left tower and $y$ is height (in meters) above the roadway. To find the cable's lowest point for roadway clearance, what is the vertex form and its practical meaning?
    Answer: $y = 0.1(x - 10)^2 + 5$; lowest point is 5 meters above roadway, 10 meters from left tower When you encounter a quadratic function in standard form like $y = ax^2 + bx + c$, converting to vertex form $y = a(x - h)^2 + k$ reveals the parabola's vertex at $(h, k)$. For real-world applications like bridge design, the vertex represents the maximum or minimum point—crucial for engineering specifications. To convert $y = 0.1x^2 - 2x + 15$ to vertex form, complete the square. Factor out the coefficient of $x^2$ from the first two terms: $y = 0.1(x^2 - 20x) + 15$. To complete the square inside the parentheses, take half of the $x$-coefficient ($-20 ÷ 2 = -10$), square it ($(-10)^2 = 100$), then add and subtract: $y = 0.1(x^2 - 20x + 100 - 100) + 15 = 0.1((x - 10)^2 - 100) + 15$. Distribute: $y = 0.1(x - 10)^2 - 10 + 15 = 0.1(x - 10)^2 + 5$. The vertex is at $(10, 5)$, meaning the cable's lowest point is 5 meters above the roadway, located 10 meters from the left tower. This confirms answer B. Answer A incorrectly shows $(x + 10)^2$, placing the vertex at $x = -10$ (10 meters before the left tower). Answer C uses the original constant term 15 instead of the correct vertex $y$-coordinate 5. Answer D miscalculates the $x$-coordinate as 20 instead of 10. Remember: when completing the square, the vertex form directly gives you $(h, k)$ from $y = a(x - h)^2 + k$, making it invaluable for optimization problems in engineering contexts.
17. A ball is thrown upward from a platform. Its height $h$ (in feet) above the ground after $t$ seconds is given by $h(t) = -16t^2 + 32t + 48$. To find when the ball reaches its maximum height, a student begins completing the square. What is the correct vertex form, and what does the vertex represent in this context?
    Answer: $h(t) = -16(t - 1)^2 + 64$; the ball reaches a maximum height of 64 feet after 1 second To complete the square for $h(t) = -16t^2 + 32t + 48$, factor out -16 from the first two terms: $h(t) = -16(t^2 - 2t) + 48$. To complete the square inside the parentheses, take half of the coefficient of $t$ and square it: $(-2/2)^2 = 1$. Add and subtract this inside: $h(t) = -16(t^2 - 2t + 1 - 1) + 48 = -16((t-1)^2 - 1) + 48 = -16(t-1)^2 + 16 + 48 = -16(t-1)^2 + 64$. The vertex $(1, 64)$ means the maximum height of 64 feet occurs at $t = 1$ second. Choice B has the wrong sign in the vertex form. Choice C incorrectly calculates the maximum height. Choice D uses the wrong time value.
18. The cost $C$ (in dollars) to produce $n$ items per day at a small factory is $C(n) = 0.5n^2 - 20n + 250$. The manager wants to find the production level that minimizes cost per day. What does the vertex form tell us about optimal production?
    Answer: $C(n) = 0.5(n - 20)^2 + 50$; produce 20 items for minimum cost of $50 per day When you encounter a quadratic cost function, you're looking at an optimization problem where the vertex form reveals the minimum or maximum point. Since the coefficient of $n^2$ is positive (0.5), this parabola opens upward, meaning the vertex represents the minimum cost. To convert $C(n) = 0.5n^2 - 20n + 250$ to vertex form, you need to complete the square. First, factor out the coefficient of $n^2$ from the first two terms: $C(n) = 0.5(n^2 - 40n) + 250$. To complete the square inside the parentheses, take half of the coefficient of $n$ (which is -40), square it: $(-20)^2 = 400$. Add and subtract this inside: $C(n) = 0.5(n^2 - 40n + 400 - 400) + 250 = 0.5((n - 20)^2 - 400) + 250 = 0.5(n - 20)^2 - 200 + 250 = 0.5(n - 20)^2 + 50$. This confirms answer C is correct: produce 20 items for a minimum cost of $50. Answer A uses the wrong constant term (250 instead of 50), likely from forgetting to distribute the 0.5 through the -400. Answer B has the wrong sign in the vertex form ($n + 20$ instead of $n - 20$), which would incorrectly place the minimum at $n = -20$. Answer D shows a calculation error in finding the vertex's x-coordinate, using 10 instead of 20. Always remember: in vertex form $a(x - h)^2 + k$, the vertex is at $(h, k)$, and when $a > 0$, this point represents the minimum value of the function.
19. The path of a diving board's end can be modeled by $y = x^2 - 8x + 19$, where $x$ is the horizontal distance (in feet) from the pool's edge and $y$ is the height (in feet) above water. To find the lowest point of the dive, what is the vertex form and its interpretation?
    Answer: $y = (x - 4)^2 + 3$; lowest point is 3 feet above water, 4 feet from pool edge When you encounter a quadratic function and need to find its vertex (maximum or minimum point), you need to convert from standard form to vertex form by completing the square. Starting with $y = x^2 - 8x + 19$, you complete the square on the $x$ terms. Take half of the coefficient of $x$ (which is $-8$), so half is $-4$. Square this to get $16$. Now rewrite: $y = x^2 - 8x + 16 + 19 - 16 = (x - 4)^2 + 3$. The vertex form $y = (x - h)^2 + k$ immediately tells you the vertex is at point $(h, k)$. Since the coefficient of $(x - 4)^2$ is positive (it's 1), the parabola opens upward, making this vertex the minimum point. So the lowest point is at $(4, 3)$, meaning 4 feet from the pool edge and 3 feet above water. Choice A incorrectly completes the square—you can verify that $(x - 8)^2 + 19$ doesn't equal the original expression when expanded. Choice B has the wrong sign in the squared term; $(x + 4)^2$ would mean the vertex is at $x = -4$, which doesn't make sense for this problem setup. Choice C correctly identifies the $x$-coordinate but gives the wrong $y$-coordinate—it uses 19 instead of 3, suggesting the student didn't properly subtract the 16 when completing the square. Remember: when completing the square, whatever you add inside the parentheses must be subtracted outside to keep the equation balanced. The vertex form immediately reveals the turning point without additional calculation.
20. Two students are completing the square for $f(x) = x^2 - 6x + 13$. Student A claims the vertex form is $f(x) = (x - 3)^2 + 4$, while Student B claims it's $f(x) = (x - 3)^2 + 13$. Which student is correct, and what mistake did the other student make?
    Answer: Student A is correct; Student B forgot to subtract the squared term when completing the square When you encounter completing the square problems, you're converting a quadratic from standard form to vertex form by creating a perfect square trinomial. Let's work through $f(x) = x^2 - 6x + 13$ step by step. To complete the square for $x^2 - 6x$, you take half the coefficient of $x$ (which is $-6/2 = -3$) and square it: $(-3)^2 = 9$. Now you can rewrite the expression as: $f(x) = x^2 - 6x + 9 - 9 + 13$ The first three terms form a perfect square: $(x - 3)^2$. The remaining terms combine to give: $-9 + 13 = 4$. Therefore, $f(x) = (x - 3)^2 + 4$. Student A is correct with this answer, confirming that choice C is right. Looking at the wrong answers: Choice A gives $f(x) = (x - 6)^2 + 13$, which incorrectly uses the full coefficient of $x$ instead of half of it when forming the perfect square. Choice B claims Student B is correct, but Student B's answer $f(x) = (x - 3)^2 + 13$ shows they forgot the crucial step of subtracting the 9 they added when completing the square. Choice D gives $f(x) = (x + 3)^2 + 4$, which has the wrong sign inside the parentheses. Remember: when completing the square, you must subtract whatever you add to maintain equality. The most common mistake is forgetting this compensation step, leading to an incorrect constant term in the final vertex form.

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